Question: The base of a triangular piece of paper $ABC$ is $12\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $16\%$ that of the area of the triangle $ABC.$ What is the length of $DE,$ in cm?

[asy]
draw((0,0)--(12,0)--(9.36,3.3)--(1.32,3.3)--cycle,black+linewidth(1));
draw((1.32,3.3)--(4,-3.4)--(9.36,3.3),black+linewidth(1));
draw((1.32,3.3)--(4,10)--(9.36,3.3),black+linewidth(1)+dashed);
draw((0,-5)--(4,-5),black+linewidth(1));
draw((8,-5)--(12,-5),black+linewidth(1));
draw((0,-4.75)--(0,-5.25),black+linewidth(1));
draw((12,-4.75)--(12,-5.25),black+linewidth(1));
label("12 cm",(6,-5));
label("$A$",(0,0),SW);
label("$D$",(1.32,3.3),NW);
label("$C$",(4,10),N);
label("$E$",(9.36,3.3),NE);
label("$B$",(12,0),SE);
[/asy]
Explanation: Let $X$ and $Y$ be the points where the folded portion of the triangle crosses $AB,$ and $Z$ be the location of the original vertex $C$ after folding.

[asy]
draw((0,0)--(12,0)--(9.36,3.3)--(1.32,3.3)--cycle,black+linewidth(1));
draw((1.32,3.3)--(4,-3.4)--(9.36,3.3),black+linewidth(1));
draw((1.32,3.3)--(4,10)--(9.36,3.3),black+linewidth(1)+dashed);
draw((0,-5)--(4,-5),black+linewidth(1));
draw((8,-5)--(12,-5),black+linewidth(1));
draw((0,-4.75)--(0,-5.25),black+linewidth(1));
draw((12,-4.75)--(12,-5.25),black+linewidth(1));
label("12 cm",(6,-5));
label("$A$",(0,0),SW);
label("$D$",(1.32,3.3),NW);
label("$C$",(4,10),N);
label("$E$",(9.36,3.3),NE);
label("$B$",(12,0),SE);
label("$X$",(2.64,0),SW);
label("$Y$",(6.72,0),SE);
label("$Z$",(4,-3.4),W);
[/asy]

We are told that the area of $\triangle XYZ$ is $16\%$ that of the area of $\triangle ABC.$

Now $\triangle ACB$ is similar to $\triangle XZY,$ since $\angle XZY$ is the folded over version of $\angle ACB$ and since $$\angle XYZ=\angle EYB =\angle DEY = \angle CED = \angle CBA$$by parallel lines and folds. Since $\triangle XZY$ is similar to $\triangle ACB$ and its area is $0.16=(0.4)^2$ that of $\triangle ACB,$ the sides of $\triangle XZY$ are $0.4$ times as long as the sides of $\triangle ACB.$

Draw the altitude of $\triangle ACB$ from $C$ down to $P$ on $AB$ (crossing $DE$ at $Q$) and extend it through to $Z.$

[asy]
draw((0,0)--(12,0)--(9.36,3.3)--(1.32,3.3)--cycle,black+linewidth(1));
draw((1.32,3.3)--(4,-3.4)--(9.36,3.3),black+linewidth(1));
draw((1.32,3.3)--(4,10)--(9.36,3.3),black+linewidth(1)+dashed);
draw((0,-5)--(4,-5),black+linewidth(1));
draw((8,-5)--(12,-5),black+linewidth(1));
draw((0,-4.75)--(0,-5.25),black+linewidth(1));
draw((12,-4.75)--(12,-5.25),black+linewidth(1));
label("12 cm",(6,-5));
label("$A$",(0,0),SW);
label("$D$",(1.32,3.3),NW);
label("$C$",(4,10),N);
label("$E$",(9.36,3.3),NE);
label("$B$",(12,0),SE);
label("$X$",(2.64,0),SW);
label("$Y$",(6.72,0),SE);
label("$Z$",(4,-3.4),W);
draw((4,10)--(4,-3.4),black+linewidth(1));
label("$Q$",(4,3.3),NE);
label("$P$",(4,0),NE);
[/asy]

Now $CP=CQ+QP=ZQ+QP=ZP+2PQ.$

Since the sides of $\triangle XZY$ are $0.4$ times as long as the sides of $\triangle ACB,$ then $ZP=0.4CP.$

Since $CP=ZP+2PQ,$ we have $PQ=0.3CP,$ and so $CQ=CP-PQ=0.7CP.$

Since $CQ$ is $0.7$ times the length of $CP,$ then $DE$ is $0.7$ times the length of $AB,$ again by similar triangles, so $DE=0.7(12)=\boxed{8.4}\text{ cm}.$